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C5×D3237 #44569162

order32370
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3237
unambiguous semidirect productC3⋊(C5×D1079)
unambiguous semidirect productC13⋊(C5×D249)
unambiguous semidirect productC83⋊(C5×D39)

Splittings

Fitting, socle, HallC3×C5×C13×C83 ⋊ C2
derived, HallC3×C13×C83 ⋊ C2×C5
HallC5×C13×C83 ⋊ D3
HallC3×C5×C83 ⋊ D13
HallC13×C83 ⋊ C5×D3
HallC5×C83 ⋊ D39
HallC3×C83 ⋊ C5×D13
HallC3×C5×C13 ⋊ D83
SylowC83 ⋊ C5×D39
HallC5×C13 ⋊ D249
HallC3×C13 ⋊ C5×D83
HallC3×C5 ⋊ D1079
SylowC13 ⋊ C5×D249
SylowC3 ⋊ C5×D1079

Subgroups

subgroupG/H
center, hypercenterC5D3237
commutator, cosocle, hypocenterC3×C13×C83C2×C5
FrattiniC1G
Fitting, socleC3×C5×C13×C83C2
LCSG ▷ C3×C13×C83
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 104
transitive degreeμt(G) = 16185

A minimal faithful permutation representation — 2 generators on 104 points:

(1,2,3,4,5)(6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104);(7,8)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Conjugacy classes & character table

8100 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32370, 13);